Matched Z-transform method
The Matched Z transform (english matched z- transform, referred to in German as adjusted Z- transformation and also as a pole -zero figure) is in signal processing, a transformation - a conversion type in Mathematics - between the continuous-time and discrete-time representation of system functions. She plays in the digital signal processing and control theory a role, since it allows an implementation to the system or between analog, digital and continuous systems, discrete systems. Transformations of a similar scope are the bilinear transformation and the Impulsinvarianzmethode.
Motivation
In the signal processing and control engineering is to implement in the field of digital signal processing the request, given continuous-time transfer functions G (s) to be implemented by linear, time-invariant systems in discrete-time transfer function H [ z] with the same behavior as possible. The transfer function G (s), for example, describe an analog filter H and [Z] is a derivative of the analog filters, discrete-time transfer function, which describes an equivalent digital filter.
The Matched Z transform offers over similar methods such as the bilinear transform the advantage of non-linear distortion of the transfer functions of G (s) to H [z ] to avoid the transmission range. Disadvantages are the caused by this method aliasing in the discrete-time system.
Description
The description of the system functions of the continuous-time systems is in the so-called S- layer, in the complex plane, and their analysis by means of Laplace transform. Wherein the discrete-time systems, the display is in the so-called z-plane and the analysis is carried out by means of the Z-transform. For linear time-invariant systems - these systems are provided in the application of the matched Z-transform - can be the transfer function G (s) as a rational function written as:
With zeroes and poles. The factor is a gain factor dar. The matched- Z transform, all poles and zeros of the relationship
Implemented. The parameter T represents the temporal sampling interval ( period ) of the discrete-time talent system, which discrete-time transfer function H [ z] is formed thus:
Graphically the transformation process in the adjacent figure is sketched qualitatively in a transmission system with three zero and three poles, the transition of each point is represented by the color match. Lie in the continuous-time system, not all poles and zeros in the s-plane within the designated value as the main strip unshaded area occurs in the discrete-time system to aliasing.
The gain in the discrete-time system is selected by comparison of characteristic frequency points. For example, in a low-pass filter for the DC component at the frequency 0 s -1. To ensure equal delay times in the time-discrete filter, in addition, it may be necessary to add more poles or zeros at.